Entropy condition for balance law of Burgers type

Question

I am studying the wellposedness of the Burgers equation with initial data:

\begin{align}\label{eq:BP} \begin{cases} \partial_t u + u \partial_x u = 0, & t >0, \quad x \in\mathbb{R}, \\ u(0,x) = u_0(x), & x \in \mathbb{R}, \end{cases} \end{align}

The definition of the classic entropy condition for the initial value problem of this equation says that this due satisfies the following two assumptions:

• (Integral equation) $$ \int_{0}^{\infty} \int_{0}^{\infty}\left(u \varphi_{t}+\frac{u^{2}}{2} \varphi_{x}\right) \mathrm{d} x \mathrm{~d} t+\left.\int_{0}^{\infty} u_{0} \varphi \mathrm{d} x\right|_{t=0}=0 $$ for all test function.
• (Entropy condition) $$ u(x+z, t)-u(x, t) \leqslant C\left(1+\frac{1}{t}\right) z $$ for some $C\geq 0$ and almost all $x,\,z\in \mathbb{R}^+,\, t>0$.

However, if now I consider \begin{align} \begin{cases} \partial_t u + u \partial_x u = G(u), & t >0, \quad x \in\mathbb{R}, \\ u(0,x) = u_0(x), & x \in \mathbb{R}, \end{cases} \end{align} How can I define the entropy condition in this case ? Is it the same that the case before? Why?

Answer 1

A good starting point may be chapter 5 of Dafermos' book, where I quote from the abstract:

It will be shown that when the system of balance laws is endowed with a companion balance law induced by a convex entropy, the initial value problem is locally well-posed in the context of classical solutions: sufficiently smooth initial data generate a classical solution defined on a maximal time interval, typically of finite duration. However, in the presence of damping induced by relaxation or other dissipative mechanisms, and when the initial data are sufficiently small, the classical solution exists globally in time. Classical solutions are unique and depend continuously on their initial values, not only within the class of classical solutions but even within the broader class of weak solutions that satisfy the companion balance law as an inequality admissibility constraint.

This deals with classical solutions first. Later weak solutions are also discussed. You can read on companion balance laws in Chapter 1.4, 1.5 of the aforementioned book.